Quantum error correction scheme for fully-correlated noise

This paper investigates quantum error correction schemes for fully-correlated noise channels on an n-qubit system, where error operators take the form \(W^{\otimes n}\), with W being an arbitrary \(2\times 2\) unitary operator. In previous literature, a recursive quantum error correction scheme can be used to protect k qubits using \((k+1)\)-qubit ancilla. We implement this scheme on 3-qubit and 5-qubit channels using the IBM quantum computers, where we uncover an error in the previous paper related to the decomposition of the encoding/decoding operator into elementary quantum gates. Here, we present a modified encoding/decoding operator that can be efficiently decomposed into (a) standard gates available in the qiskit library and (b) basic gates comprised of single-qubit gates and CNOT gates. Since IBM quantum computers perform relatively better with fewer basic gates, a more efficient decomposition gives more accurate results. Our experiments highlight the importance of an efficient decomposition for the encoding/decoding operators and demonstrate the effectiveness of our proposed schemes in correcting quantum errors. Furthermore, we explore a special type of channel with error operators of the form \(\sigma _x^{\otimes n}, \sigma _y^{\otimes n}\) and \(\sigma _z^{\otimes n}\), where \(\sigma _x, \sigma _y, \sigma _z\) are the Pauli matrices. For these channels, we implement a hybrid quantum error correction scheme that protects both quantum and classical information using IBM’s quantum computers. We conduct experiments for \(n = 3, 4, 5\) and show significant improvements compared to recent work.